The first part of the paper describes the circumstances under which tidal energy supplied to the atmosphere through the action of tide-producing forces can be trapped between a certain stratum (usually where the temperature has a minimum) and the ground. The results are then applied to discuss in general terms the types of free oscillation which an atmosphere with a given temperature distribution may possess. It is pointed out that Kelvin's hypothesis that the atmosphere has a resonance in the neighbourhood of 12 solar hours leads directly to the conclusion that the temperature must fall again to a low value at some level above the hot region inferred from observations of the anomalous propagation of sound. In the second part of the paper results are given of numerical calculations made with the aid of a differential analyzer to determine to what extent the requirements of the resonance theory restrict the possible temperature variation in the atmosphere. The results of Appleton & Weekes (1939) on lunar tides in the E region are discussed, and it is shown that there is no difficulty in reconciling them with oscillation theory provided a suitable temperature distribution in the E region is assumed.